Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.9 AIME Find the answer

In triangle $A B C$ , points $M$ and $N$ are the midpoints of $A B$ and $A C$ , respectively, and points $P$ and $Q$ trisect $B C$ . Given that $A, M, N, P$ , and $Q$ lie on a circle and $B C=1$ , compute the area of triangle $A B C$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Note that $M P \parallel A Q$ , so $A M P Q$ is an isosceles trapezoid. In particular, we have $A M=M B=B P=P Q=\frac{1}{3}$ , so $A B=\frac{2}{3}$ . Thus $A B C$ is isosceles with base 1 and legs $\frac{2}{3}$ , and the height from $A$ to $B C$ is $\frac{\sqrt{7}}{6}$ , so the area is $\frac{\sqrt{7}}{12}$ .

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